Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a submarine.
Answer the following questions:
a) Both ships are aiming at the same submarine, and both fire a missile. What is the probability that the submarine sinks.
b) Given that the submarine is seen sinking, what is the probability that both missiles hit.
Attempt:
(a) $P(A \cup B) = P(A) + P(B) – P(A\cap B) = P(A) + P(B) – P(A) P(B) = 1.3 – 0.4 = 0.9$
(b) $P ( A \cap B \mid A \cup B) = \dfrac{P(A\cap B \cap (A\cup B) )}{P(A \cup B)}=\dfrac{P(A \cap B)}{P(A)+P(B)-P(A)P(B)} = \dfrac{4}{9}$
Are these correct? Im confused how to treat it when $P(A)+P(B) >1$
Best Answer
It is correct if $A$ and $B$ are independent. The fact that $P(A)+P(B)>1$ doesn't upset anything. But see my comments on notation above.